Limit of $\frac{x^{4} - 1}{x - 1}$ as $x \to 1$

Q: What is the limit of (x^4 - 1) / (x - 1) as x approaches 1?

The limit of x^4 - 1x - 1 as x → 1 is 4.

The limit of $\frac{x^{4} - 1}{x - 1}$ as $x \to 1$ is $4$ .

\lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = 4

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Graph of (x^4 - 1) / (x - 1) near x = 1. The open circle marks the point x = 1, where the function may be undefined; the curve still heads to 4.

How to work it out

1
We want what ( $x^{4}$ - 1) / (x - 1) approaches as x → 1.
$\lim_{x \to 1} \frac{x^{4} - 1}{x - 1}$
2
Direct substitution gives an indeterminate form: 0/0. It does not tell us the limit.
3
L'Hôpital's rule: when the form is 0/0, the limit equals the limit of the ratio of the derivatives.
$\lim \frac{x^{4} - 1}{x - 1} = \lim \frac{4 x^{3}}{1}$
4
Now substitute x = 1: the denominator is not 0 any more.
$\frac{4}{1} = 4$
5
Method: this is the derivative of $x^{4}$ at 1, namely 4.
6
Result:
$\lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = 4$

LaTeX source of the formulas on this page

\[ \lim_{x \to 1} \frac{x^{4} - 1}{x - 1} = 4 \]
\[ \lim_{x \to 1} \frac{x^{4} - 1}{x - 1} \]
\[ \lim \frac{x^{4} - 1}{x - 1} = \lim \frac{4 x^{3}}{1} \]
\[ \frac{4}{1} = 4 \]

Values of ( $x^{4}$ - 1) / (x - 1) as x → 1 (computed)

x	f(x)	Distance from the limit
1.1	4.6410000	6.41e-1
0.9	3.4390000	5.61e-1
1.01	4.0604010	6.04e-2
0.99	3.9403990	5.96e-2
1.001	4.0060040	6.00e-3
0.999	3.9940040	6.00e-3
1.0001	4.0006000	6.00e-4
0.9999	3.9994000	6.00e-4
1.00001	4.0000600	6.00e-5
0.99999	3.9999400	6.00e-5

Questions

What is the limit of (

x^{4}

- 1) / (x - 1) as x approaches 1?

The limit of $\frac{x^{4} - 1}{x - 1}$ as $x \to 1$ is $4$ .

Which form is it?

0/0.

Limit of x4−1x−1\frac{x^{4} - 1}{x - 1}x−1x4−1​ as x→1x \to 1x→1

How to work it out

Values of (x4x^{4}x4 - 1) / (x - 1) as x → 1 (computed)

Questions

More well-known limits

Limit of $\frac{x^{4} - 1}{x - 1}$ as $x \to 1$

Values of ( $x^{4}$ - 1) / (x - 1) as x → 1 (computed)